Arc-length control method (Lam & Morley, 1992)

Contents

Notation and references

The notation followed here and in the following MATLAB codes:

conforms to that used by Lam & Morley in the following reference:

Lam, W. and Morley, C. (1992). ”Arc-Length Method for Passing Limit Points in Structural Calculation.” J. Struct. Eng., 118(1), 169–185. This reference is denoted as [6] inside the text of the above codes.

Algorithms implemented

  1. Arc length control method as described by Lam & Morley (1992)
  2. Modified version of the above method which directs the search towards $$\mathrm{\lambda}=1$ , where $$\mathrm{\lambda}$ is the load factor.
help arc_length_Lam_Morley %1
help arc_length_Lam_Morley_modified %2

  Arc-length control method (Lam & Morley, 1992)
 
  Description
      The equation functn(#p#)=#lambda#*#q# is solved for #p#, where
      #lambda# (load factor) is given as a result along with each root
      (#p#), with #q# not equal to 0. If the equation functn(#p#)=0 is to
      be solved, #q# and #functn# are redefined such that
      functn(#p#)+#q#=#q#. 
      The method used is described in Lam & Morley (1992): "Arc-Length
      Method for Passing Limit Points in Structural Calculation" according
      to the flowchart of Figure 3, p.178.
      Notation in this code conforms to that used in the above paper.
 
  Required input arguments
      #functn# is the function handle at the left hand side of the equation
      to be solved. The definition of #functn# must be of the type:
      [#f#,#J#]=functn(#p#), where #f# ([#dim# x 1]) is the value of
      #functn# at #p# ([#dim# x 1]) and #J# ([#dim# x #dim#]) is the value
      of the Jacobian matrix of #functn# at #p#.
      #q# ([#dim# x 1]) is the right hand side of the equation
      functn(#p#)=#lambda#*#q#.
      #p0# ([#dim# x 1]) is the starting point of the solution.
 
  Optional input arguments
      #maxIINCS# (scalar) is the number of equilibrium points desired.
      Default value is 10.
      #lambda0# (scalar) is the initial load factor. Default value
      is 0.
      #Deltalambda# (scalar) is the initial load increment. Default value
      is 1.
      #IITERmax# (scalar) is the maximum number of iterations permitted for
      each converged point. If convergence is not achieved until the
      maximum iteration the procedure stops and accepts as equilibrium
      point that calculated at the last iteration. Default value is 20, as
      recommended by Lam & Morley (1992).
      #Ide# (scalar) is the desired number of iterations to each converged
      point. Default value is 1.
      #tol# (scalar) is the tolerance for convergence criterion (eq. 20) in
      [6] for #e# and #h#. Default value is 0.00005.
      #KTup# (scalar) is the stiffness matrix updater (number of iterations
      after which the tangent stiffness matrix is updated). For #KTup# = 1
      the algorithm implemented is Full Arc-Length method. For #KTup# = Inf
      the algorithm implemented is Initial Stiffness Arc-Length method.
      Default value is 1.
      #dettol# (scalar) is the tolerance for singularity of Jacobian (#J#).
      Default value is 1e-4.
 
  Output arguments
      #p_out# ([#dim# x #maxIINCS#]) roots of the equation being solved.
      #lambda_out# ([1 x #maxIINCS#]) are the load factors (one per
      increment). The roots of the equation to be solved correspond to
      #lambda_out#=1 (or a feasible value as closest as possible to 1).
      #iter_out# ([1 x #maxIINCS#]) number of iterations for each
      increment.
 
  Parents (calling functions)
      None.
 
  Children (called functions)
      None.
 
 __________________________________________________________________________
  Copyright (c) 09-Mar-2014
      George Papazafeiropoulos
      First Lieutenant, Infrastructure Engineer, Hellenic Air Force
      Civil Engineer, M.Sc., Ph.D. candidate, NTUA
      Email: gpapazafeiropoulos@yahoo.gr
      Website: http://users.ntua.gr/gpapazaf/
 
 

  Modified arc-length control method (Lam & Morley, 1992)
 
  Description
      The equation functn(#p#)=#lambda#*#q# is solved for #p#, where
      #lambda# (load factor) is given as a result along with each root
      (#p#), with #q# not equal to 0. If the equation functn(#p#)=0 is to
      be solved, #q# and #functn# are redefined such that
      functn(#p#)+#q#=#q#. 
      The method used is described in Lam & Morley (1992): "Arc-Length
      Method for Passing Limit Points in Structural Calculation" according
      to the flowchart of Figure 3, p.178, with a minor modification so
      that the solution procedure is directed towards #lambda#=1.
      Notation in this code conforms to that used in the above paper.
 
  Required input arguments
      #functn# is the function handle at the left hand side of the equation
      to be solved. The definition of #functn# must be of the type:
      [#f#,#J#]=functn(#p#), where #f# ([#dim# x 1]) is the value of
      #functn# at #p# ([#dim# x 1]) and #J# ([#dim# x #dim#]) is the value
      of the Jacobian matrix of #functn# at #p#.
      #q# ([#dim# x 1]) is the right hand side of the equation
      functn(#p#)=#lambda#*#q#.
      #p0# ([#dim# x 1]) is the starting point of the solution.
 
  Optional input arguments
      #maxIINCS# (scalar) is the number of equilibrium points desired.
      Default value is 10.
      #lambda0# (scalar) is the initial load factor. Default value
      is 0.
      #Deltalambda# (scalar) is the initial load increment. Default value
      is 1.
      #IITERmax# (scalar) is the maximum number of iterations permitted for
      each converged point. If convergence is not achieved until the
      maximum iteration the procedure stops and accepts as equilibrium
      point that calculated at the last iteration. Default value is 20, as
      recommended by Lam & Morley (1992).
      #Ide# (scalar) is the desired number of iterations to each converged
      point. Default value is 1.
      #tol# (scalar) is the tolerance for convergence criterion (eq. 20) in
      [6] for #e# and #h#. Default value is 0.00005.
      #KTup# (scalar) is the stiffness matrix updater (number of iterations
      after which the tangent stiffness matrix is updated). For #KTup# = 1
      the algorithm implemented is Full Arc-Length method. For #KTup# = Inf
      the algorithm implemented is Initial Stiffness Arc-Length method.
      Default value is 1.
      #dettol# (scalar) is the tolerance for singularity of Jacobian (#J#).
      Default value is 1e-4.
 
  Output arguments
      #p_out# ([#dim# x #maxIINCS#]) roots of the equation being solved.
      #lambda_out# ([1 x #maxIINCS#]) are the load factors (one per
      increment). The roots of the equation to be solved correspond to
      #lambda_out#=1 (or a feasible value as closest as possible to 1).
      #iter_out# ([1 x #maxIINCS#]) number of iterations for each
      increment.
 
  Parents (calling functions)
      None.
 
  Children (called functions)
      None.
 
 __________________________________________________________________________
  Copyright (c) 09-Mar-2014
      George Papazafeiropoulos
      First Lieutenant, Infrastructure Engineer, Hellenic Air Force
      Civil Engineer, M.Sc., Ph.D. candidate, NTUA
      Email: gpapazafeiropoulos@yahoo.gr
      Website: http://users.ntua.gr/gpapazaf/

Equations solved

The following equations are solved for $$\mathrm{x_i}$ and $$\mathrm{\lambda}$

$$x^3 - \frac{57\, x^2}{8} + \frac{51\, x}{4} = 5\, \mathrm{\lambda} \ \ \ \ \ \ (1)$$

$$\left[\begin{array}{c} {\mathrm{x_1}}^2 + {\mathrm{x_2}}^2 - 49\\ \mathrm{x_1}\, \mathrm{x_2} - 24 \end{array}\right] = \left[\begin{array}{c} 1\\ 1 \end{array}\right] \, \mathrm{\lambda} \ \ \ \ \ \ (2)$$

Function definitions

Two functions are utilized for the arc-length procedure:

The first function ($f_1$, defined in the file function4.m ), needed to solve equation (1) is a cubic polynomial with the following properties:

$$f_1\left( x \right) = x^3 - \frac{57\, x^2}{8} + \frac{51\, x}{4}$$

$$J_1\left( x \right) = 3\, x^2 - \frac{57\, x}{4} + \frac{51}{4} $$

$$f_1\left( 0 \right) = 0 $$

The second function ($f_2$, defined in the file function3.m ), needed to solve equation (2) is a nonlinear smooth function with the following properties:

$$f_2\left(\left[\begin{array}{c} \mathrm{x_1}\\ \mathrm{x_2} \end{array}\right]\right) = \left[\begin{array}{c} {\mathrm{x_1}}^2 + {\mathrm{x_2}}^2 - 49\\ \mathrm{x_1}\, \mathrm{x_2} - 24 \end{array}\right]$$

$$J_2\left(\left[\begin{array}{c} \mathrm{x_1}\\ \mathrm{x_2} \end{array}\right]\right) = \left[\begin{array}{cc} 2\, \mathrm{x_1} & 2\, \mathrm{x_2}\\ \mathrm{x_2} & \mathrm{x_1} \end{array}\right] $$

Function coding

function [f,J]=function4(x)
% Function output (1-by-1)
f=x^3-57/8*x^2+51/4*x;
% Function Jacobian output (1-by-1)
J=3*x^2-57/4*x+51/4;
end

function [f,J]=function3(x)
% Function output column vector (2-by-1)
f1 = x(1)^2 + x(2)^2 - 49;
f2 = x(1)*x(2) -24;
f = [f1;f2];
% Function Jacobian output matrix (2-by-2)
J=[ 2*x(1), 2*x(2);
    x(2),   x(1)];
end

Initial definitions

In the subsequent code the following initial definitions are made (in the order presented below):

  1. Define function $f_1$
  2. Define function $f_2$
  3. Set right hand side ($q$) of equation (1)
  4. Set right hand side ($q$) of equation (2)
  5. Set starting point ($p_0$) for solution of equation (1)
  6. Set starting point ($p_0$) for solution of equation (2)
  7. Set number of increments desired
  8. Set initial value of $$\mathrm{\lambda_0}$
  9. Set initial value of $$\mathrm{\Delta\lambda}$
  10. Set maximum number of iterations permitted per increment
  11. Set number of iterations desired to each converged point ($$I^{de}$)
  12. Set number of iterations desired to each converged point ($$I^{de}$) for the modified Lam-Morley algorithm and solution of equation (2)
  13. Set tolerance for convergence ($$\mathrm{\epsilon}$) for the solution of equation (1). Typical values range from 1/1000 to 1/500
  14. Set tolerance for convergence ($$\mathrm{\epsilon}$) for the solution of equation (2). Typical values range from 1/1000 to 1/500
  15. Set the number of iterations every which the stiffness matrix of the problem is updated. KTup=1 corresponds to the full arc length method
  16. Set the tolerance for determining if the stiffness matrix is singular (this is true if its determinant is below dettol)
functn1=@function4; %1
functn2=@function3; %2
q1=5; %3
q2=[1;1]; %4
p01=0.1; %5
p02=[4;6]; %6
maxIINCS=10; %7
lambda0=0; %8
Deltalambda=1; %9
IITERmax=20; %10
Ide=1; %11
Idemod2=5; %12
tol=5e-5; %13
tol2=2e-3; %14
KTup=1; %15
dettol=1e-4; %16

Applications

  1. Default application of the arc length control method as described by Lam & Morley (1992) to solve equation (1)
  2. Default application of the modified version of the Lam & Morley (1992) arc length control method to solve equation (1)
  3. Non-default application of the arc length control method as described by Lam & Morley (1992) to solve equation (1)
  4. Non-default application of the modified version of the Lam & Morley (1992) arc length control method to solve equation (1)
  5. Default application of the arc length control method as described by Lam & Morley (1992) to solve equation (2)
  6. Default application of the modified version of the Lam & Morley (1992) arc length control method to solve equation (2)
  7. Non-default application of the arc length control method as described by Lam & Morley (1992) to solve equation (2)
  8. Non-default application of the modified version of the Lam & Morley (1992) arc length control method to solve equation (2)
[p_out1,lambda_out1,iter_out1] = arc_length_Lam_Morley(functn1,q1,p01); %1
Result1=[p_out1',lambda_out1',iter_out1'] %1
[p_out2,lambda_out2,iter_out2] = arc_length_Lam_Morley_modified(functn1,q1,p01); %2
Result2=[p_out2',lambda_out2',iter_out2'] %2
[p_out3,lambda_out3,iter_out3] = arc_length_Lam_Morley(functn1,q1,p01,maxIINCS,lambda0,Deltalambda,IITERmax,Ide,tol,KTup,dettol); %3
Result3=[p_out3',lambda_out3',iter_out3'] %3
[p_out4,lambda_out4,iter_out4] = arc_length_Lam_Morley_modified(functn1,q1,p01,maxIINCS,lambda0,Deltalambda,IITERmax,Ide,tol,KTup,dettol); %4
Result4=[p_out4',lambda_out4',iter_out4'] %4
[p_out5,lambda_out5,iter_out5] = arc_length_Lam_Morley(functn2,q2,p02); %5
Result5=[p_out5',lambda_out5',iter_out5'] %5
[p_out6,lambda_out6,iter_out6] = arc_length_Lam_Morley_modified(functn2,q2,p02); %6
Result6=[p_out6',lambda_out6',iter_out6'] %6
[p_out7,lambda_out7,iter_out7] = arc_length_Lam_Morley(functn2,q2,p02,maxIINCS,lambda0,Deltalambda,IITERmax,Ide,tol2,KTup,dettol); %7
Result7=[p_out7',lambda_out7',iter_out7'] %7
[p_out8,lambda_out8,iter_out8] = arc_length_Lam_Morley_modified(functn2,q2,p02,maxIINCS,lambda0,Deltalambda,IITERmax,Idemod2,tol2,KTup,dettol); %8
Result8=[p_out8',lambda_out8',iter_out8'] %8

Result1 =

    0.5403    0.9934    2.0000
    0.9807    1.3189    2.0000
    1.2522    1.3514    2.0000
    1.4420    1.3137    2.0000
    1.5824    1.2594    2.0000
    1.6590    1.2217    2.0000
    1.7300    1.1822    2.0000
    1.7636    1.1621    2.0000
    1.7992    1.1399    2.0000
    1.8149    1.1298    2.0000


Result2 =

    0.5403    0.9934    2.0000
    0.9807    1.3189    2.0000
    1.2522    1.3514    2.0000
    1.4420    1.3137    2.0000
    1.5824    1.2594    2.0000
    1.6590    1.2217    2.0000
    1.7300    1.1822    2.0000
    1.7636    1.1621    2.0000
    1.7992    1.1399    2.0000
    1.8149    1.1298    2.0000


Result3 =

    0.5403    0.9934    2.0000
    0.9807    1.3189    2.0000
    1.2522    1.3514    2.0000
    1.4420    1.3137    2.0000
    1.5824    1.2594    2.0000
    1.6590    1.2217    2.0000
    1.7300    1.1822    2.0000
    1.7636    1.1621    2.0000
    1.7992    1.1399    2.0000
    1.8149    1.1298    2.0000


Result4 =

    0.5403    0.9934    2.0000
    0.9807    1.3189    2.0000
    1.2522    1.3514    2.0000
    1.4420    1.3137    2.0000
    1.5824    1.2594    2.0000
    1.6590    1.2217    2.0000
    1.7300    1.1822    2.0000
    1.7636    1.1621    2.0000
    1.7992    1.1399    2.0000
    1.8149    1.1298    2.0000


Result5 =

    5.8220    3.7946   -1.3961   20.0000
    5.5474    4.0874   -1.3812   20.0000
    5.5973    4.0344   -1.3859   20.0000
    5.5888    4.0483   -1.3851   20.0000
    5.5888    4.0489   -1.3714    2.0000
    5.5888    4.0489   -1.3713    2.0000
    5.5888    4.0489   -1.3713    2.0000
    5.5888    4.0489   -1.3712    1.0000
    5.5888    4.0489   -1.3712    2.0000
    5.5888    4.0489   -1.3713    1.0000


Result6 =

    5.3339    4.8149    2.1037   20.0000
    5.0730    5.0229    1.8003   20.0000
    5.0697    5.0263    1.7341   20.0000
    5.0700    5.0276    1.7193   20.0000
    5.0687    5.0292    1.7224   20.0000
    5.0684    5.0296    1.7232   20.0000
    5.0683    5.0296    1.7234   20.0000
    5.0683    5.0296    1.7234   20.0000
    5.0683    5.0297    1.7234   20.0000
    5.0683    5.0297    1.7234   20.0000


Result7 =

    5.8220    3.7946   -1.3961   20.0000
    5.5928    4.0375   -1.4193   11.0000
    5.5939    4.0345   -1.4312    3.0000
    5.5936    4.0352   -1.4289    3.0000
    5.5946    4.0347   -1.4252    2.0000
    5.5938    4.0356   -1.4251    2.0000
    5.5930    4.0364   -1.4257    1.0000
    5.5922    4.0372   -1.4264    1.0000
    5.5926    4.0367   -1.4264    2.0000
    5.5934    4.0357   -1.4260    1.0000


Result8 =

    5.3339    4.8149    2.1037   20.0000
    5.0388    4.9887    1.3831   20.0000
    5.0226    4.9892    1.1042   20.0000
    5.0137    4.9870    1.0052    6.0000
    5.0136    4.9869    1.0038    2.0000
    5.0134    4.9867    1.0006    2.0000
    5.0131    4.9865    0.9971    2.0000
    5.0828    4.9136    0.9763   15.0000
    5.0596    4.9375    0.9803    2.0000
    5.0044    4.9959    1.0022    3.0000

Copyright

Copyright (c) 09-Mar-2014 by George Papazafeiropoulos


Published with MATLAB® R2012b